Statistical Estimation

1. Front Matter

   Pages i-vii

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2. Basic Notation

   • I. A. Ibragimov, R. Z. Has’minskii

   Pages 1-2

3. Introduction

   • I. A. Ibragimov, R. Z. Has’minskii

   Pages 3-9

4. The Problem of Statistical Estimation

   • I. A. Ibragimov, R. Z. Has’minskii

   Pages 10-112

5. Local Asymptotic Normality of Families of Distributions

   • I. A. Ibragimov, R. Z. Has’minskii

   Pages 113-172

6. Properties of Estimators in the Regular Case

   • I. A. Ibragimov, R. Z. Has’minskii

   Pages 173-213

7. Some Applications to Nonparametric Estimation

   • I. A. Ibragimov, R. Z. Has’minskii

   Pages 214-240

8. Independent Identically Distributed Observations. Densities with Jumps

   • I. A. Ibragimov, R. Z. Has’minskii

   Pages 241-280

9. Independent Identically Distributed Observations. Classification of Singularities

   • I. A. Ibragimov, R. Z. Has’minskii

   Pages 281-320

10. Several Estimation Problems in a Gaussian White Noise

    • I. A. Ibragimov, R. Z. Has’minskii

    Pages 321-361

11. Back Matter

    Pages 363-403

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when certain parameters in the problem tend to limiting values (for example, when the sample size increases indefinitely, the intensity of the noise ap­ proaches zero, etc.) To address the problem of asymptotically optimal estimators consider the following important case. Let X 1, X 2, ... , X n be independent observations with the joint probability density !(x,O) (with respect to the Lebesgue measure on the real line) which depends on the unknown patameter o e 9 c R1. It is required to derive the best (asymptotically) estimator 0:( X b ... , X n) of the parameter O. The first question which arises in connection with this problem is how to compare different estimators or, equivalently, how to assess their quality, in terms of the mean square deviation from the parameter or perhaps in some other way. The presently accepted approach to this problem, resulting from A. Wald's contributions, is as follows: introduce a nonnegative function w(0l> ( ), Ob Oe 9 (the loss function) and given two estimators Of and O! n 2 2 the estimator for which the expected loss (risk) Eown(Oj, 0), j = 1 or 2, is smallest is called the better with respect to Wn at point 0 (here EoO is the expectation evaluated under the assumption that the true value of the parameter is 0). Obviously, such a method of comparison is not without its defects.

• Asymptotische Wirksamkeit
• Estimation Theory
• Estimator
• Parameter
• Schätzung (Statistik)

• LOMI, Leningrad, USSR

  I. A. Ibragimov

• Doz., Institut Problem Peredači Inf., Moscow, USSR

  R. Z. Has’minskii